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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"486047421498a75936df79748dc70257dd60e5b3f321d4aba7966c8c126eba2d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a9d89adf599d215a","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:             1;\n  1:            35,             16;\n  2:           969,           1120,            256;\n  3:         24187,          46512,          26880,           4096;\n  4:        569297,        1547968,        1488384,         573440,           65536;\n  5:      12901779,       45543760,       61918720,       39690240,        11468800,        1048576;\n  6:     284751385,     1238570784,     2186100480,     1981399040,       952565760,      220200960,       16777216;\n  7:    6163010411,    31892155120,    69359963904,    81614417920,     55479173120,    21337473024,     4110417920,     268435456;\n  8:  131399145633,   788865332608,  2041097927680,  2959358459904,   2611661373440,  1420266831872,   455199424512,   75161927680,    4294967296;\n  9: 2768320775683, 18921476971152, 56798303947776, 97972700528640, 106536904556544, 75215847555072, 34086403964928, 9364102447104, 1352914698240, 68719476736;\n","recurrence_sentence":"The row polynomials satisfy (16*n+3+16*x)*P(n,x) = (304*n+361+(256*n+608)*x+256*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((304*n + 361 + (256*n + 608)*x + 256*x^2)*P[n - 1, x])/(16*n + 3 + 16*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((304*n + 361 + (256*n + 608)*x + 256*x^2)*v[i - 1])/(16*n + 3 + 16*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2f02b2035f8f4541","steps":["basis A013609","D+1"],"edges":["e-6db2596ebcec1df1","e-6be7068733455262"],"text":"A038322 \u2192 basis A013609 \u2192 D+1"},"headline":"A038322 \u2192 basis A013609 \u2192 D+1","chain":null}